Finite Group Databasebeta

← #11739168

Subgroups — #11739168

Order 896.

332 conjugacy classes of subgroups.

The subgroup lattice has 332 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#90410C211yesmaximal, characteristic
2#90380C211yesmaximal, characteristic, complemented
3C7×#490C211yesmaximal, characteristic, complemented
4#89842C211yesmaximal, characteristic, complemented
5#89836C211yesmaximal, characteristic, complemented
6#89556C211yesmaximal, characteristic, complemented
7#89551C211yesmaximal, characteristic, complemented
8#89549C211yesmaximal, characteristic, complemented
9#89546C211yesmaximal, characteristic, complemented
10#89538C211yesmaximal, characteristic, complemented
11#89535C211yesmaximal, characteristic, complemented
12#89533C211yesmaximal, characteristic, complemented
13#89531C211yesmaximal, characteristic, complemented
14#89481C211yesmaximal, characteristic, complemented
15#89474C211yesmaximal, characteristic, complemented
16C7×#127C2211yescharacteristic
17C7×#125C2211yescharacteristic
18#6502—21no—
19#6502—21no—
20#6501C2211yescharacteristic
21#6501C2211yescharacteristic
22#6498—21no—
23#6498—21no—
24D4×Dic7C2211yescharacteristic
25D4×Dic7C2211yescharacteristic
26#6460—21no—
27#6459—21no—
28#6455C2211yescharacteristic
29D7×C4⋊C4C2211yescharacteristic
30#6453C2211yescharacteristic
31#6452—21no—
32#6451—21no—
33#6451C2211yescharacteristic
34#6435C2211yescharacteristic
35#6433C2211yescharacteristic
36#6432C2211yescharacteristic
37C7×#102C2211yescharacteristic
38C7×#102C2211yescharacteristic
39C7×#102C2211yescharacteristic
40C7×#102C2211yescharacteristic
41C7×#97C2211yescharacteristic
42#6406C2211yescharacteristic
43#6406C2211yescharacteristic
44#6406C2211yescharacteristic
45#6406C2211yescharacteristic
46(C2×D7)⋊C8C2211yescharacteristic
47(C2×D7)⋊C8C2211yescharacteristic
48#6392C2211yescharacteristic
49#6392C2211yescharacteristic
50#6391C2211yescharacteristic
51#6391C2211yescharacteristic
52Dic7⋊C8C2211yescharacteristic
53Dic7⋊C8C2211yescharacteristic
54#6382C2211yescharacteristic
55#6382C2211yescharacteristic
56#6381C2211yescharacteristic
57#6381C2211yescharacteristic
58C4×C7⋊C8C2211yescharacteristic
59#3241—71nomaximal, Sylow
60C2×C7×D4—21no—
61C2×C7×D4—21no—
62C2×C7×D4C2311yescharacteristic
63C2×C7×D4C2311yescharacteristic
64C2×C22⋊D7—21no—
65C2×C22⋊D7—21no—
66C22×Dic7—21no—
67C22×Dic7—21no—
68C2×C4×D7—21no—
69C2×C4×D7C2311yescharacteristic
70C2×Dic14—21no—
71C2×Dic14—21no—
72C2×C7×C8C2311yescharacteristic
73C2×C7×C8C2311yescharacteristic
74C7×C4⋊C4—21no—
75C7×C4⋊C4—21no—
76C7×C4⋊C4C2311yescharacteristic
77C7×C4⋊C4C2311yescharacteristic
78C42×C7C2311yescharacteristic
79C22⋊Dic7—21no—
80C22⋊Dic7—21no—
81C22⋊Dic7—21no—
82C22⋊Dic7—21no—
83(C2×D7)⋊C4—21no—
84(C2×D7)⋊C4D411yescharacteristic
85(C2×D7)⋊C4D411yescharacteristic
86C4⋊Dic7—21no—
87C4⋊Dic7—21no—
88C4⋊Dic7C2311yescharacteristic
89C4⋊Dic7C2311yescharacteristic
90C4⋊Dic7C2311yescharacteristic
91C4⋊Dic7C2311yescharacteristic
92Dic7⋊C4—21no—
93Dic7⋊C4—21no—
94Dic7⋊C4—21no—
95Dic7⋊C4—21no—
96Dic7⋊C4—21no—
97Dic7⋊C4D411yescharacteristic
98Dic7⋊C4D411yescharacteristic
99C4×Dic7—21no—
100C4×Dic7C2311yescharacteristic
101C2×C7⋊C8C2311yescharacteristic
102C2×C7⋊C8C2311yescharacteristic
103#544—71no—
104#540—71no—
105#501—71no—
106#499—71no—
107#490D711yescharacteristic, complemented
108#486—71no—
109#482—71no—
110#482—71no—
111#481—71no—
112#480—71no—
113#478—71no—
114#478—71no—
115#476—71no—
116#474—71no—
117#433—71no—
118C23×C7—21no—
119C23×C7—21no—
120C22×D7C2×D411yescharacteristic
121C7×D4—21no—
122C7×D4—21no—
123C7×D4—21no—
124C7×D4—21no—
125C7×D4—81no—
126C7×D4—81no—
127C2×C4×C7—21no—
128C2×C4×C7—21no—
129C2×C4×C7C2×D411yescharacteristic
130C2×C4×C7C2×D411yescharacteristic
131C2×C4×C7C2411yescharacteristic
132C22⋊D7—81no—
133C22⋊D7—81no—
134C2×Dic7—21no—
135C2×Dic7—21no—
136C2×Dic7—21no—
137C2×Dic7—21no—
138C2×Dic7—81no—
139C2×Dic7—81no—
140C2×Dic7C2×D411yescharacteristic
141C2×Dic7C2×D411yescharacteristic
142C2×Dic7C2×D411yescharacteristic
143C4×D7—21no—
144C4×D7—21no—
145C4×D7—81no—
146Dic14—81no—
147Dic14—81no—
148C7×C8—21no—
149C7×C8—21no—
150C7⋊C8—21no—
151C7⋊C8—21no—
152C4⋊Q8—71no—
153C4⋊Q8—71no—
154C4⋊Q8—141no—
155#127C2×D711yescharacteristic
156#125—71no—
157#125—71no—
158#125—141no—
159#125C2×D711yescharacteristic
160#123—141no—
161#123—141no—
162C22⋊Q8—141no—
163C22⋊Q8—141no—
164#121—71no—
165#121—71no—
166#121—141no—
167#121—141no—
168C4×D4—71no—
169C4×D4—71no—
170#117—71no—
171#117—71no—
172C2×C4⋊C4—71no—
173#107—71no—
174#107—71no—
175#107—71no—
176#107—71no—
177#106—71no—
178#106—71no—
179#106—71no—
180#106—71no—
181C4⋊C8—71no—
182C4⋊C8—71no—
183#102—71no—
184#102—71no—
185#102—71no—
186#102—71no—
187#102C2×D711yescharacteristic
188#102C2×D711yescharacteristic
189#102C2×D711yescharacteristic
190#102C2×D711yescharacteristic
191C22⋊C8—71no—
192C22⋊C8—71no—
193#97C2×D711yescharacteristic
194C4×C8—71no—
195C22×C7—41no—
196C22×C7—41no—
197C22×C7—41no—
198C22×C7—41no—
199C22×C7—41no—
200C22×C7—41no—
201C22×C7C22×D411yescharacteristic
202C2×D7—21no—
203C2×D7—21no—
204C2×D7—21no—
205C4×C7—21no—
206C4×C7—21no—
207C4×C7—41no—
208C4×C7—41no—
209C4×C72−1+411yescharacteristic
210C4×C72−1+411yescharacteristic
211Dic7—21no—
212Dic7—21no—
213Dic7—21no—
214Dic7—41no—
215Dic7—41no—
216Dic7—41no—
217Dic7—41no—
218C2×Q8—141no—
219C2×Q8—141no—
220C2×D4—21no—
221C2×D4—21no—
222C2×D4—141no—
223C2×D4—141no—
224C2×D4C22×D711yescharacteristic
225C2×D4C22×D711yescharacteristic
226C22×C4—71no—
227C22×C4—141no—
228C22×C4—141no—
229C22×C4—141no—
230C2×C8—71no—
231C2×C8—71no—
232C2×C8C22×D711yescharacteristic
233C2×C8C22×D711yescharacteristic
234C4⋊C4—21no—
235C4⋊C4—21no—
236C4⋊C4—71no—
237C4⋊C4—71no—
238C4⋊C4—71no—
239C4⋊C4—71no—
240C4⋊C4—71no—
241C4⋊C4—71no—
242C4⋊C4—141no—
243C4⋊C4—141no—
244C4⋊C4—141no—
245C4⋊C4—141no—
246C4⋊C4—141no—
247C4⋊C4—141no—
248C4⋊C4—141no—
249C4⋊C4C22×D711yescharacteristic
250C4⋊C4C22×D711yescharacteristic
251C22⋊C4—71no—
252C22⋊C4—71no—
253C22⋊C4—141no—
254C22⋊C4—141no—
255C22⋊C4—141no—
256C22⋊C4—141no—
257C22⋊C4—141no—
258C42—71no—
259C42—141no—
260C42C22×D711yescharacteristic
261C2×C7—81no—
262C2×C7—81no—
263C2×C7#53611yescharacteristic
264C2×C7#57611yescharacteristic
265C2×C7#57711yescharacteristic
266D7—41no—
267C23—21no—
268C23—21no—
269C23—71no—
270Q8—561no—
271Q8—561no—
272D4—21no—
273D4—21no—
274D4—21no—
275D4—21no—
276D4—81no—
277D4—81no—
278D4—561no—
279D4—561no—
280C2×C4—21no—
281C2×C4—21no—
282C2×C4—71no—
283C2×C4—71no—
284C2×C4—71no—
285C2×C4—141no—
286C2×C4—141no—
287C2×C4—141no—
288C2×C4—141no—
289C2×C4—141no—
290C2×C4—141no—
291C2×C4—561no—
292C2×C4—561no—
293C2×C4—561no—
294C2×C4D4×D711yescharacteristic
295C2×C4D4×D711yescharacteristic
296C2×C4C23×D711yescharacteristic
297C8—21no—
298C8—21no—
299C8—141no—
300C8—141no—
301C7#324111yescharacteristic, complemented, Sylow
302C22—41no—
303C22—41no—
304C22—41no—
305C22—41no—
306C22—41no—
307C22—41no—
308C22—141no—
309C22—141no—
310C22—141no—
311C22C2×D4×D711yescharacteristic
312C4—21no—
313C4—21no—
314C4—41no—
315C4—41no—
316C4—141no—
317C4—141no—
318C4—141no—
319C4—281no—
320C4—281no—
321C4—281no—
322C4—281no—
323C4#655411yescharacteristic
324C4#655411yescharacteristic
325C2—81no—
326C2—81no—
327C2—281no—
328C2#9036611yescharacteristic
329C2#9046511yescharacteristic
330C2#9046611yescharacteristic
subgroupG/H
centerC22C2×D4×D7
commutator, cosocleC2×C4×C7C24
FrattiniC2×C4C23×D7
FittingC7×#490C2
socleC22×C7C22×D4
prime radical#90410C2
hypercenter#490D7
hypocenterC7#3241
LCSG ▷ C2×C4×C7 ▷ C2×C7 ▷ C7
UCSC1 ◁ C22 ◁ C42 ◁ #490

Complements

Splittings G = N ⋊ H with N normal; H ≅ G/N.

NH ≅ G/Ntype
2. #90380C2semidirect
3. C7×#490C2semidirect
4. #89842C2semidirect
5. #89836C2semidirect
6. #89556C2semidirect
7. #89551C2semidirect
8. #89549C2semidirect
9. #89546C2semidirect
10. #89538C2semidirect
11. #89535C2semidirect
12. #89533C2semidirect
13. #89531C2semidirect
14. #89481C2semidirect
15. #89474C2semidirect
107. #490D7semidirect
301. C7#3241semidirect

Maximal subgroups

By isomorphism type — maximal subgroups of the same type appear once, whether or not they are normal.

subgroupindex
#904102
#903802
C7×#4902
#898422
#898362
#895562
#895512
#895492
#895462
#895382
#895352
#895332
#895312
#894812
#894742
#32417